In this paper we give optimal lower bounds for the blow-up rate of the 𝐻˙𝑠(𝕋3)-norm, 1/2<𝑠<5/2, of a putative singular solution of the Navier–Stokes equations, and we also present an elementary proof for a lower bound on blow-up rate of the Sobolev norms of possible singular solutions to the Euler equations when 𝑠>52